Existence, uniqueness and homogenization of nonlinear parabolic problems with dynamical boundary conditions in perforated media

Abstract

We consider a nonlinear parabolic problem with nonlinear dynamical boundary conditions of pure-reactive type in a media perforated by periodically distributed holes of size $\varepsilon$. The novelty of our work is to consider a nonlinear model where the nonlinearity also appears in the boundary. The existence and uniqueness of solution is analyzed. Moreover, passing to the limit when $\varepsilon$ goes to zero, a new nonlinear parabolic problem defined on a unified domain without holes with zero Dirichlet boundary condition and with extra-terms coming from the influence of the nonlinear dynamical boundary conditions is rigorously derived.